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Flat knot 6.1685

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,2,2,3,0,0,2,1,1,2,2,0,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.1685']
Arrow polynomial of the knot is: -4*K1*K2 + 2*K1 + 2*K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.540', '6.925', '6.1021', '6.1117', '6.1120', '6.1135', '6.1227', '6.1230', '6.1260', '6.1682', '6.1685', '6.1922']
Outer characteristic polynomial of the knot is: t^7+48t^5+111t^3+12t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1685']
2-strand cable arrow polynomial of the knot is: 128*K1**4*K2 - 1584*K1**4 + 512*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1216*K1**3*K3 - 864*K1**2*K2**2 - 320*K1**2*K2*K4 + 4224*K1**2*K2 - 1488*K1**2*K3**2 - 64*K1**2*K3*K5 - 48*K1**2*K4**2 - 3648*K1**2 + 64*K1*K2**3*K3 - 96*K1*K2**2*K3 - 96*K1*K2*K3*K4 + 4264*K1*K2*K3 + 1912*K1*K3*K4 + 192*K1*K4*K5 - 32*K2**4 - 64*K2**2*K3**2 - 16*K2**2*K4**2 + 400*K2**2*K4 - 2660*K2**2 + 200*K2*K3*K5 + 32*K2*K4*K6 - 1900*K3**2 - 780*K4**2 - 140*K5**2 - 12*K6**2 + 3066
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1685']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16987', 'vk6.17229', 'vk6.20215', 'vk6.21505', 'vk6.23392', 'vk6.23701', 'vk6.27407', 'vk6.29023', 'vk6.35448', 'vk6.35890', 'vk6.38824', 'vk6.41013', 'vk6.42885', 'vk6.43187', 'vk6.45585', 'vk6.47352', 'vk6.55144', 'vk6.55394', 'vk6.57054', 'vk6.58174', 'vk6.59519', 'vk6.59870', 'vk6.61567', 'vk6.62739', 'vk6.64960', 'vk6.65167', 'vk6.66673', 'vk6.67511', 'vk6.68253', 'vk6.68410', 'vk6.69325', 'vk6.70078']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is c.
The reverse -K is
The mirror image K* is
The reversed mirror image -K* is
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

invariant value
Gauss code O1O2O3U4O5U2U1U6O4O6U3U5
R3 orbit {'O1O2O3U4O5U2U1U6O4O6U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U1O5O6U5U3U2O4U6
Gauss code of K* O1O2O3U4U3O5O6U2U1U5O4U6
Gauss code of -K* O1O2O3U4O5U6U3U2O4O6U1U5
Diagrammatic symmetry type c
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -1 -1 1 -2 2 1],[ 1 0 0 1 0 2 2],[ 1 0 0 0 1 1 2],[-1 -1 0 0 -2 0 0],[ 2 0 -1 2 0 3 2],[-2 -2 -1 0 -3 0 -2],[-1 -2 -2 0 -2 2 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -2 -1 -2 -3],[-1 0 0 0 0 -1 -2],[-1 2 0 0 -2 -2 -2],[ 1 1 0 2 0 0 1],[ 1 2 1 2 0 0 0],[ 2 3 2 2 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,0,2,1,2,3,0,0,1,2,2,2,2,0,-1,0]
Phi over symmetry [-2,-1,-1,1,1,2,-1,0,2,2,3,0,0,2,1,1,2,2,0,0,2]
Phi of -K [-2,-1,-1,1,1,2,1,2,1,1,1,0,0,1,1,0,2,2,0,-1,1]
Phi of K* [-2,-1,-1,1,1,2,-1,1,1,2,1,0,0,0,1,1,2,1,0,1,2]
Phi of -K* [-2,-1,-1,1,1,2,-1,0,2,2,3,0,0,2,1,1,2,2,0,0,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial -2w^4z^2+7w^3z^2-2w^3z+24w^2z+25w
Inner characteristic polynomial t^6+36t^4+71t^2+4
Outer characteristic polynomial t^7+48t^5+111t^3+12t
Flat arrow polynomial -4*K1*K2 + 2*K1 + 2*K3 + 1
2-strand cable arrow polynomial 128*K1**4*K2 - 1584*K1**4 + 512*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1216*K1**3*K3 - 864*K1**2*K2**2 - 320*K1**2*K2*K4 + 4224*K1**2*K2 - 1488*K1**2*K3**2 - 64*K1**2*K3*K5 - 48*K1**2*K4**2 - 3648*K1**2 + 64*K1*K2**3*K3 - 96*K1*K2**2*K3 - 96*K1*K2*K3*K4 + 4264*K1*K2*K3 + 1912*K1*K3*K4 + 192*K1*K4*K5 - 32*K2**4 - 64*K2**2*K3**2 - 16*K2**2*K4**2 + 400*K2**2*K4 - 2660*K2**2 + 200*K2*K3*K5 + 32*K2*K4*K6 - 1900*K3**2 - 780*K4**2 - 140*K5**2 - 12*K6**2 + 3066
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}]]
If K is slice False
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